Download presentation
Presentation is loading. Please wait.
Published byMildred Murphy Modified over 9 years ago
1
15-251 Great Theoretical Ideas in Computer Science
2
Algebraic Structures: Group Theory Lecture 15 (October 12, 2010)
3
Number Theory Naturals closed under + a+b = b+a a+0 = 0+a Integers closed under + a+b = b+a a+0 = 0+a a+(-a) = 0 ZnZn closed under + n a+ n b = b+ n a a+ n 0 = 0+ n a a+ n (-a) = 0 (a+b)+c = a+(b+c) (a+ n b)+ n c = a+ n (b+ n c)
4
A+0 = 0+A a+0 = 0+a a+(-a) = 0 a+ n 0 = 0+ n a a+ n (-a) = 0 closed under * closed under * n closed under * (a+b)*c = a*c+b*c ditto A+(-A) = 0 1/a may not exist ditto Number Theory closed under + A+B = B+A closed under + a+b = b+a closed under + n a+ n b = b+ n a (a+b)+c = a+(b+c) (A+B)+C = A+(B+C) Matrices Integers ZnZn (a+ n b)+ n c = a+ n (b+ n c)
5
A+0 = 0+A a+0 = 0+a a+(-a) = 0 a+ n 0 = 0+ n a a+ n (-a) = 0 closed under * closed under * n closed under * (a+b)*c = a*c+b*c ditto A+(-A) = 0 1/a exists if a 0 ditto Number Theory closed under + A+B = B+A closed under + a+b = b+a closed under + n a+ n b = b+ n a (a+b)+c = a+(b+c) (a+ n b)+ n c = a+ n (b+ n c) (A+B)+C = A+(B+C) Invertible Matrices Rationals Z n (n prime)
6
Abstraction: Abstract away the inessential features of a problem =
7
Today we are going to study the abstract properties of binary operations
8
Rotating a Square in Space Imagine we can pick up the square, rotate it in any way we want, and then put it back on the white frame
9
In how many different ways can we put the square back on the frame? R 90 R 180 R 270 R0R0 F|F| F—F— FF We will now study these 8 motions, called symmetries of the square
10
Symmetries of the Square Y SQ = { R 0, R 90, R 180, R 270, F |, F —, F, F }
11
Composition Define the operation “ ” to mean “first do one symmetry, and then do the next” For example, R 90 R 180 Question: if a,b Y SQ, does a b Y SQ ? Yes! means “first rotate 90˚ clockwise and then 180˚” = R 270 F | R 90 means “first flip horizontally and then rotate 90˚” = F
12
R 90 R 180 R 270 R0R0 F|F| F—F— FF R0R0 R 90 R 180 R 270 F|F| F—F— F F R0R0 R 90 R 180 R 270 F|F| F—F— FF R 90 R 180 R 270 F|F| F—F— F F R 180 R 270 R0R0 R0R0 R 90 R0R0 R 180 FFF|F| F—F— F—F— F|F| FF FFF—F— F|F| FF—F— F FF|F| F F—F— FF|F| F|F| FF—F— R0R0 R0R0 R0R0 R0R0 R 90 R 270 R 180 R 270 R 90 R 270 R 90 R 180 R 90 R 270 R 180
13
How many symmetries for n-sided body?2n R 0, R 1, R 2, …, R n-1 F 0, F 1, F 2, …, F n-1 R i R j = R i+j R i F j = F j-i F j R i = F j+i F i F j = R j-i
14
Some Formalism If S is a set, S S is: the set of all (ordered) pairs of elements of S S S = { (a,b) | a S and b S } If S has n elements, how many elements does S S have? n2n2 Formally, is a function from Y SQ Y SQ to Y SQ : Y SQ Y SQ → Y SQ As shorthand, we write (a,b) as “a b”
15
“ ” is called a binary operation on Y SQ Definition: A binary operation on a set S is a function : S S → S Example: The function f: → defined by is a binary operation on f(x,y) = xy + y Binary Operations
16
Is the operation on the set of symmetries of the square associative? A binary operation on a set S is associative if: for all a,b,c S, (a b) c = a (b c) Associativity Examples: Is f: → defined by f(x,y) = xy + y associative? (ab + b)c + c = a(bc + c) + (bc + c)?NO! YES!
17
A binary operation on a set S is commutative if For all a,b S, a b = b a Commutativity Is the operation on the set of symmetries of the square commutative? NO! R 90 F | ≠ F | R 90
18
R 0 is like a null motion Is this true: a Y SQ, a R 0 = R 0 a = a? R 0 is called the identity of on Y SQ In general, for any binary operation on a set S, an element e S such that for all a S, e a = a e = a is called an identity of on S Identities YES!
19
Inverses Definition: The inverse of an element a Y SQ is an element b such that: a b = b a = R 0 Examples: R 90 inverse: R 270 R 180 inverse: R 180 F|F| inverse: F |
20
Every element in Y SQ has a unique inverse
21
R 90 R 180 R 270 R0R0 F|F| F—F— FF R0R0 R 90 R 180 R 270 F|F| F—F— F F R0R0 R 90 R 180 R 270 F|F| F—F— FF R 90 R 180 R 270 F|F| F—F— F F R 180 R 270 R0R0 R0R0 R 90 R0R0 R 180 FFF|F| F—F— F—F— F|F| FF FFF—F— F|F| FF—F— F FF|F| F F—F— FF|F| F|F| FF—F— R0R0 R0R0 R0R0 R0R0 R 90 R 270 R 180 R 270 R 90 R 270 R 90 R 180 R 90 R 270 R 180
22
3. (Inverses) For every a S there is b S such that: Groups A group G is a pair (S, ), where S is a set and is a binary operation on S such that: 1. is associative 2. (Identity) There exists an element e S such that: e a = a e = a, for all a S a b = b a = e
23
Commutative or “Abelian” Groups remember, “commutative” means a b = b a for all a, b in S If G = (S, ) and is commutative, then G is called a commutative group
24
To check “group-ness” Given (S, ) 1.Check “closure” for (S, ) (i.e, for any a, b in S, check a b also in S). 2.Check that associativity holds. 3.Check there is a identity 4.Check every element has an inverse
25
Some examples…
26
Examples Is ( ,+) a group? Is + associative on ? YES! Is there an identity?YES: 0 Does every element have an inverse?NO! ( ,+) is NOT a group Is closed under +? YES!
27
Examples Is (Z,+) a group? Is + associative on Z? YES! Is there an identity?YES: 0 Does every element have an inverse?YES! (Z,+) is a group Is Z closed under +? YES!
28
Examples Is (Odds,+) a group? (Odds,+) is NOT a group Is + associative on Odds? YES! Is there an identity?NO! Does every element have an inverse?YES! Is Odds closed under +? NO!
29
Examples Is (Y SQ, ) a group? Is associative on Y SQ ? YES! Is there an identity?YES: R 0 Does every element have an inverse?YES! (Y SQ, ) is a group the “dihedral” group D 4 Is Y SQ closed under ? YES!
30
Examples Is (Z n,+ n ) a group? Is + n associative on Z n ? YES! Is there an identity?YES: 0 Does every element have an inverse?YES! (Z n, + n ) is a group (Z n is the set of integers modulo n) Is Z n closed under + n ? YES!
31
Examples Is (Z n,* n ) a group? Is * n associative on Z n ? YES! Is there an identity?YES: 1 Does every element have an inverse?NO! (Z n, * n ) is NOT a group (Z n is the set of integers modulo n)
32
Examples Is (Z n *, * n ) a group? Is * n associative on Z n * ? YES! Is there an identity?YES: 1 Does every element have an inverse? YES! (Z n *, * n ) is a group (Z n * is the set of integers modulo n that are relatively prime to n)
33
(Z, *) (Q, *) the rationals (Q \ {0}, *) No inverses… Zero has no inverse… Yes
34
3. (Inverses) For every a S there is b S such that: Groups A group G is a pair (S, ), where S is a set and is a binary operation on S such that: 1. is associative 2. (Identity) There exists an element e S such that: e a = a e = a, for all a S a b = b a = e
35
Some properties of groups…
36
Theorem: A group has at most one identity element Proof: Suppose e and f are both identities of G=(S, ) Then f = e f = e Identity Is Unique We denote this identity by “e” exactly one identity
37
Theorem: Every element in a group has a unique inverse Proof: Inverses Are Unique Suppose b and c are both inverses of a Then b = b e = b (a c) = (b a) c = c
38
Theorem: If a b = a c, then b = c Proof: Cancellation
39
Orders and generators
40
A group G=(S, ) is finite if S is a finite set Define |G| = |S| to be the order of the group (i.e. the number of elements in the group) What is the group with the least number of elements? How many groups of order 2 are there? G = ({e}, ) where e e = e e f ef e f f e Order of a group
41
e a b e a b
42
Generators A set T S is said to generate the group G = (S, ) if every element of S can be expressed as a finite “sum” of elements in T Question: Does {R 90 } generate Y SQ ? Question: Does {F |, R 90 } generate Y SQ ? An element g S is called a generator of G=(S, ) if the set {g} generates G Does Y SQ have a generator? NO! YES! NO!
43
Generators For (Z n,+) Any a Z n such that GCD(a,n)=1 generates (Z n,+) Claim: If GCD(a,n) =1, then the numbers a, 2a, …, (n-1)a, na are all distinct modulo n Proof (by contradiction): Suppose xa = ya (mod n) for x,y {1,…,n} and x ≠ y Then n | a(x-y) Since GCD(a,n) = 1, then n | (x-y), which cannot happen
44
If G = (S, ), we use a t denote (a a … a) t times If G = (Z n, +), this means “ a t ” denotes (a + a + … + a) Order of an element If G = (Z n *, *), “ a t ” now denotes a t mod n Please be careful when using notation “ a t ” ! Warning: Potential Confusion = t*a mod n t times
45
If G = (S, ), we use a t denote (a a … a) t times Definition: The order of an element a of G is the smallest positive integer n such that a n = e The order of an element can be infinite! Example: The order of 1 in the group (Z,+) is infinite What is the order of F | in Y SQ ?2 What is the order of R 90 in Y SQ ?4 Order of an element
46
Remember order of a group G = size of the group G order of an element g in group G = (smallest n>0 s.t. g n = e)
47
Theorem: If G is a finite group, then for all g in G, order(g) is finite. Orders Proof: Consider g, g g, g g g = g 3, g 4, … Since G is finite, g j = g k for some j < k g j = g j g k-j e = g k-j Multiplying both sides by (g j ) -1
48
Remember order of a group G = size of the group G order of an element g = (smallest n>0 s.t. g n = e) g is a generator of group G if order(g) = order(G)
49
What is order(Z n, + n )? Orders n For x in (Z n, + n ), what is order(x)? order(x) = n/GCD(x,n)
50
Orders order(Z n *, * n )? Á (n) For x in (Z n, * n ), what is order(x)? At most Á (n) (Euler’s theorem)
51
Theorem: Let x be an element of G. The order of x divides the order of G Orders Corollary: If p is prime, a p-1 = 1 (mod p) (remember, this is Fermat’s Little Theorem) What group did we use for the corollary? G = (Z p *, *), order(G) = p-1
52
Groups and Subgroups
53
Subgroups Suppose G = (S, ) is a group. If T µ S, and if H = (T, ) is also a group, then H is called a subgroup of G.
54
Examples (Z, +) is a group and (Evens, +) is a subgroup. In fact, (Multiples of k, +) is also a subgroup. Is (Odds, +) a subgroup of (Z,+) ? No! (Odds,+) is not even a group!
55
Examples (Z n, + n ) is a group and if k | n, Is ({0, k, 2k, 3k, …, (n/k-1)k}, + n ) subgroup of (Z n,+ n ) ? Is (Z k, + k ) a subgroup of (Z n, + n )? Is (Z k, + n ) a subgroup of (Z n, + n )? No! it doesn’t even have the same operation No! (Z k, + n ) is not a group! (not closed) Only if k is a divisor of n.
56
Subgroup facts (identity) If e is the identity in G = (S, ), what is the identity in H = (T, )? e Proof: Clearly, e satisfies But we saw there is a unique such element in any group. e a = a e = a for all a in T.
57
Subgroup facts (inverse) If b is a’s inverse in G = (S, ), what is a’s inverse in H = (T, )? b Proof: let c be a’s inverse in H. Then c a = e c a b = e b c e = b c = b
58
Lagrange’s Theorem Theorem: If G is a finite group, and H is a subgroup then the order of H divides the order of G. In symbols, |H| divides |G|. Corollary: If x in G, then order(x) divides |G|. Proof of Corollary: Consider the set T x = (x, x 2 = x x, x 3, …) H = (T x, ) is a group. (check!) Hence it is a subgroup of G = (S, ). Order(H) = order(x). (check!)
59
Lagrange’s Theorem Theorem: If G is a finite group, and H is a subgroup then the order of H divides the order of G. Curious (and super-useful) corollary: If you can show that H is a subgroup of G and H G then |H| is at most ½ |G|
60
“Right” way of looking at primality testing Fermat: if n prime, then a n-1 = 1 (mod n) for all 0 < a < n. Suppose the converse was also true: “if n composite, then exists g with 0 < g < n. g n-1 != 1 (mod n)” Then consider “bad elements” for this n: elements b such that b n-1 = 1 (mod n) Bad elements form a subgroup of Z n Picking random element, it is good with probability ½. |Bad| < ½ n Sadly, converse not true. Fixing that gives Miller-Rabin.
61
Symmetries of the Square Compositions Groups Binary Operation Identity and Inverses Basic Facts: Inverses Are Unique Generators Order of element, group Subgroups Lagrange’s theorem Here’s What You Need to Know…
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.